History
The theorem was developed at American Telephone & Telegraph as part of ongoing investigations into improved filters for telephone multiplexing applications. This work was commercially important, large sums of money could be saved by increasing the number of telephone conversations that could be carried on one line. The theorem was first published by Campbell in 1922 but without a proof. Great use was immediately made of the theorem in filter design, it appears prominently, along with a proof, in Zobel's landmark paper of 1923 which summarised the state of the art of filter design at that time. Foster published his paper the following year which included his canonical realisation forms.
Cauer in Germany grasped the importance of Foster's work and used it as the foundation of network synthesis. Amongst Cauer's many innovations was to extend Foster's work to all 2-element-kind networks after discovering an isomorphism between them. Cauer was interested in finding the conditions for realisability of a rational one-port network from its polynomial function (the condition of being a Foster network is not a necessary and sufficient condition, for that, see positive-real function) and the reverse problem of which networks were equivalent, that is, had the same polynomial function. Both of these were important problems in network theory and filter design.
Read more about this topic: Foster's Reactance Theorem
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